Theor Chem Acc (2005) 114: 283-296
DOI 10.1007/s00214-005-0681-9
R E G U L A R A RT I C L E
Kirk A Peterson · Cristina Puzzarini
Systematically convergent basis sets for transition metals.
II. Pseudopotential-based correlation consistent basis sets
for the group 11 (Cu, Ag, Au) and 12 (Zn, Cd, Hg) elements
Received: 8 June 2005 / Accepted: 14 June 2005 / Published online: 12 August 2005
© Springer-Verlag 2005
Abstract Sequences of basis sets that systematically
converge towards the complete basis set (CBS) limit have
been developed for the coinage metals (Cu, Ag, Au) and
group 12 elements (Zn, Cd, Hg). These basis sets are based
on recently published small-core relativistic pseudopotentials
[Figgen D, Rauhut G, Dolg M, Stoll H (2005) Chem Phys
311:227] and range in size from double- through quintupleζ . Series of basis sets designed for valence-only and outercore electron correlation are presented, as well as these sets
augmented by additional diffuse functions for the accurate
description of negative ions and weak interactions. Selected
benchmark calculations at the coupled cluster level of theory are presented for both atomic and molecular properties.
The latter include the calculation of both spectroscopic and
thermochemical properties of the homonuclear dimers Cu2 ,
Ag2 , and Au2 , as well as the van der Waals species Zn2 , Cd2 ,
and Hg2 . The CBS limit results, including the effects of corevalence correlation and spin-orbit coupling, represent some
of the most accurate carried out to date and result in new recommendations for the equilibrium bond lengths of the group
12 dimers. Comparisons are also made to a limited number of all-electron Douglas–Kroll–Hess (DKH) calculations
(second and third order) carried out using new correlation
consistent basis sets of triple-ζ quality.
Keywords Basis sets · Correlation · Consistent · Transition
metal dimers · Complete basis set · Coupled cluster
K.A. Peterson (B) · C. Puzzarini
Department of Chemistry,
Washington State University,
Pullman, Washington 99164-4630
E-mail: kipeters@wsu.edu
C. Puzzarini
Dipartimento di Chimica “G. Ciamician”,
Universitá di Bologna, Via Selmi 2,
40126 Bologna, Italy
E-mail: cristina.puzzarini@unibo.it
1 Introduction
With the advent of atomic natural orbital (ANO) [1] and
correlation consistent (cc) [2] basis sets, errors due to basis set
truncation effects in the solutions of the electronic Schrödinger equation could be addressed in a systematic manner. In
the particular case of correlation consistent basis sets, various extrapolation formulas have been used that can provide
accurate estimates of the complete basis set (CBS) limit for
various atomic and molecular properties, thereby removing
errors due to an incomplete basis. This then greatly facilitates
the analysis of the errors intrinsic to the chosen wavefunction
approximation, e.g., Møller–Plesset perturbation or coupled
cluster theory. Having this kind of control over the errors in a
calculation has proven to be essential in the area of ab initio
thermochemistry [3–7], where removing basis set truncation
errors is a necessary ingredient for reliably obtaining chemical accuracy (<1 kcal/mol). Until recently, however, correlation consistent basis sets have not been generally available
for the transition metal elements or even main group elements
heavier than krypton. In the case of the post- d elements Ga–
Rn, correlation consistent basis sets based on accurate relativistic pseudopotentials were only recently developed [8, 9]
and were shown to have convergence properties in molecular calculations similar to those now commonly observed for
lighter elements. In regards to the transition metal elements,
correlation consistent all-electron basis sets had been previously developed by Bauschlicher and coworkers for just two
elements, Ti and Fe [10,11]. Very recently, however, both
nonrelativistic and scalar relativistic cc-type basis sets (allelectron) from triple- to quintuple-ζ have been reported for
Sc–Zn [12].
While there are many basis set choices for the first row
transition metal atoms (Ref. [12] and references therein),
accurate basis sets for correlated calculations on the second
and third row transition metals are more scarce. Except for the
recent nonrelativistic basis sets of Osanai et al. [13] for Y–Cd
and the relativistic basis sets of Dyall [14] for Hf–Hg, these
are, with a few exceptions, limited to relatively small sets
[15–17], including those accompanying relativistic effective
284
core potentials (ECPs) [18–26]. Third-order Douglas–Kroll
optimized basis sets have also recently been reported by Hirao and coworkers [27,28], but these sets are only between
double- and triple-ζ quality.
While full families of accurate relativistic cc basis sets
could certainly also be developed for the second and third row
transition metals using, e.g., the Douglas–Kroll–Hess (DKH)
Hamiltonian [29,30] for scalar relativity, the required sizes
of the underlying Hartree-Fock (HF) primitive sets could become prohibitively large. On the other hand, small-core relativistic pseudopotentials (PPs) or effective core potentials
(ECPs) are capable of accurately recovering both scalar and
spin-orbit relativistic effects with the added benefit of removing the requirement for explicit basis functions describing
the low-lying core electrons since these are removed by the
pseudopotential. Just as the development of the PP-based
post- d correlation consistent basis sets [8,9] were motivated
by new, next-generation PPs by the Stuttgart/Köln groups [24,
25], the present work was also initiated by the adjustment
of new pseudopotentials for the group 11 and 12 elements
by these same groups [26]. This paper reports the development of correlation consistent-type basis sets for Cu–Au and
Zn–Hg based on the small core relativistic PPs of Figgen
et al. [26]. The resulting sets are denoted cc-pV nZ-PP for
valence-only correlation, cc-pwCV nZ-PP for valence plus
outer-core correlation, and aug-cc-p(wC)VnZ for their diffuse function-augmented versions. In each case nZ ranges
from DZ to 5Z and they are very similar in construction to
the previously reported basis sets for both the first row transition metals (all-electron) and the post- d elements (PP-based).
Some preliminary basis sets of this type had been previously
reported [31] for the Hg atom using the older PP of Häussermann et al. [32], and the present work supercedes those
sets.
Of course, it is still important to also determine what
additional errors have been introduced into the calculations
by using the PP approximation. To provide the best possible
comparison, all-electron basis sets of triple-ζ quality have
also been developed in this work for the second and third
row transition metal elements of this study using the DKH
Hamiltonian (cc-pVTZ-DK, cc-pwCVTZ-DK, and aug-ccp(wC)VTZ-DK). The determination of basis sets for all the
remaining elements of the transition metal series that are
equivalent to the present work, both PP-based and all-electron, is also currently underway [33].
The details of the basis set development is described in
Sect. 2.1, which is followed in Sect. 2.2 by a description of
the atomic (excitation energies, electron affinities, ionization
potentials) and molecular (spectroscopic and thermochemical properties of Cu2 , Ag2 , Au2 , Zn2 , Cd2 , Hg2 ) benchmark
calculations that have been carried out to ascertain the accuracy and efficiency of the new basis sets and underlying PPs.
The results of the benchmark calculations are presented and
discussed in Sect. 3 with conclusions given in Sect. 4.
K.A. Peterson, C. Puzzarini
2 Methodology
2.1 Basis set construction
The basis set development of the present work closely follows that of the all-electron correlation consistent basis sets
for the first row transition metals that were recently developed in this laboratory [12] (cc-pVnZ-DK and cc-pVnZ with
n = T, Q, and 5). The present cases mainly differ by the use of
small-core relativistic pseudopotentials that replace the inner
core electrons, yielding basis set series denoted as cc-pVnZPP (n = D, T, Q, and 5). Specifically the pseudopotentials
correspond to the recently reported Stuttgart/Köln energy
consistent PPs that were adjusted to all-electron four-component multi-configuration Dirac–Hartree–Fock (MCDHF)
calculations, including the two-electron Breit interaction [26].
These PPs replace the 1s–2p, 1s–3d, and 1s–4f cores of
Cu–Zn, Ag–Cd, and Au–Hg, respectively. As in our previous work, the exponents of the present basis sets were optimized employing double-sided numerical derivatives within
a BFGS algorithm [34]. The energy gradients were typically
converged to better than 1 × 10−6 . The MOLPRO program
suite [35] was used throughout and only the pure spherical harmonic components of the dfghi angular momentum
functions were utilized. All orbitals were fully symmetry
equivalenced and the frozen core approximation was imposed
throughout unless otherwise noted.
Naturally the largest difference between the present
PP-based basis sets and the all-electron ones is the decrease
in the number of primitive spd functions required for the Hartree–Fock (HF) description of the atoms due to the removal
of the low-lying core electrons from the calculations. As in
the all-electron case, the choice of HF primitive set for a particular correlation consistent basis set is dictated both by the
error in the atomic HF calculation relative to the HF limit
and the qualitative nature of the outermost exponents, e.g.,
the set chosen for the cc-pVDZ-PP basis set should provide
a double-ζ description of the ns valence orbital. The sizes
chosen for the present work, which are shown in Table 1,
were nearly identical to those employed previously for the
post-d cc-pVnZ-PP basis sets [8,9] and smoothly converge
the atomic HF energies to within a few µEh of the apparent
HF limits as the basis set is increased from DZ to 5Z. As in
the all-electron transition metal basis sets, the HF s functions
were fully optimized for the ns 2 states of the atoms, while the
p exponents were obtained from calculations on the lowest
ns2 (n − 1) d 8 np 1 and ns2 (n − 1) d 9 np 1 atomic states for the
group 11 and 12 elements, respectively. The specific states
used were the same as those given in Ref. [12] for Cu and Zn.
The optimization of the HF d functions were also carried out
in the same manner as the all-electron basis sets. Namely, the
d exponents for the coinage metal atoms were determined in
state-averaged HF calculations involving both the ns2 (n − 1)
d 9 and ns1 (n − 1) d 10 states in order to obtain a sufficiently
Systematically convergent basis sets for transition metals
285
Table 1 Composition of the basis sets (primitives)/[contracted], developed in the present work
Basis set
cc-pVDZ-PP
cc-pVTZ-PP
cc-pVQZ-PP
cc-pV5Z-PP
cc-pwCVDZ-PP
cc-pwCVTZ-PP
cc-pwCVQZ-PP
cc-pwCV5Z-PP
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
cc-pVTZ-DK
cc-pVQZ-DK
(14s11p11d3f 2g1h)/[6s6p5d3f 2g1h]a
(16s13p12d4f 3g2h1i)/[7s7p6d4f 3g2h1i]a
(20s16p8d2f 1g)/[7s6p4d2f 1g]c
(22s18p10d3f 2g1h)/[8s7p5d3f 2g1h]c
(8s7p6d1f )/[4s4p3d1f ]
(10s9p8d2f 1g)/[5s5p4d2f 1g]
(14s11p10d3f 2g1h)/[6s6p5d3f 2g1h]b
(16s13p11d4f 3g2h1i)/[7s7p6d4f 3g2h1i]b
[4s4p3d] + 1s1p1d2f
[5s5p4d] + 2s2p2d3f 2g
[6s6p5d] + 2s2p2d4f 3g2h
[7s7p6d] + 2s2p2d5f 4g3h2i
cc-pVDZ-PP + 1s1p1d1f
c-pVTZ-PP + 1s1p1d1f 1g
cc-pVQZ-PP + 1s1p1d1f 1g1h
cc-pV5Z-PP + 1s1p1d1f 1g1h1i
(26s21p14d2f 1g)/[8s7p5d2f 1g]d (32s28p19d13f 1g)/[9s8p6d3f 1g]e
(32s25p18d3f 2g1h)/[9s8p6d3f 2g1h]f
a
Cu/Zn
Ag/Cd, Au/Hg
c
Ref. [12]
d
Ag/Cd
e
Au/Hg
f
Determined for Cd only
b
diffuse set of d exponents. The analogous exponents for the
group 12 elements were only optimized for the ground ns2
(n − 1) d 10 configuration. The resulting spd HF primitive
sets were then generally contracted to [2s1p1d] using atomic
orbital (AO) coefficients from either state-averaged (s 2 and
s 1 , group 11) or single-state (s 2 , group 12) HF calculations.
An additional contracted p function was also included to
describe the valence np atomic orbital. These contraction
coefficients were obtained from HF calculations on the same
states described above for the p exponent optimizations.
The correlating functions that were added to the contracted [2s2p1d] basis sets also followed the same general
prescription as the all-electron transition metal sets [12]. The
spd correlating functions were obtained from state-averaged
(2 states for group 11, 1 state for group 12) configuration
interaction singles and doubles (CISD) atomic natural orbitals (ANOs); 1s1p1d for DZ, 2s2p2d for TZ, 3s3p3d for QZ,
and 4s4p4d for 5Z. Using ANOs rather than simply uncontracting functions from the HF sets avoids the special problems associated with pseudopotential basis sets as outlined
previously by Blaudeau et al. [36] and Christiansen [37]. In
order to provide additional flexibility, the outermost exponent
was also uncontracted from the HF set. The number and type
of higher angular momentum functions follows the expected
pattern, i.e., 1f for DZ, 2f 1g for TZ, 3f 2g1h for QZ, and
4f 3g2h1i for 5Z. These exponents were constrained to have
an even tempered distribution [38,39], i.e., ζk = αβ k , and
were optimized for the CISD energy of either a two-state
average (group 11) or the ground-state (group 12). It might
be noted that the resulting 2f 1g exponents for the cc-pVTZPP basis sets were very similar to those optimized previously
by Martin and Sundermann [40] using quasirelativistic Stuttgart-type pseudopotentials [23, 41].
Recovering the effects of core-valence electron correlation
generally requires the addition of extra correlating functions,
and cc-pwCVnZ-PP basis sets (n = D, T, Q, 5) have been
developed in this work following the same procedure as the
all-electron sets [12]. The additional functions were designed
for correlating the outer-core (n − 1) sp electrons and the
weighted core-valence prescription [42] was followed, i.e.,
the exponents were optimized for the intershell, core–valence
CISD correlation energy with the addition of just 1% of the
total intrashell, core-core correlation energy (s 2 state only in
each case). To each cc-pVnZ-PP basis set with n = T, Q, and
5, a set of uncontracted 2s2p2d functions optimized in this
manner was added. Only a 1s1p1d set of core-valence functions was added to the DZ basis set. As in the all-electron basis sets [12], the higher angular momentum functions (fghi)
were determined by retaining only the most diffuse functions from the cc-pVnZ-PP basis sets and optimizing even
tempered sequences of functions for the sum of the weighted
core-valence correlation energy and the valence correlated
CISD energy (state averaged energy for group 11). The final
composition of the resulting cc-pwCVnZ-PP basis sets are
shown in Table 1.
Finally, in order to accurately describe negative ions and
weakly bound systems, diffuse augmented basis sets were
developed. As in the all-electron transition metal work, these
additional functions, one in each angular momentum symmetry, were simply obtained as even tempered extensions of
the outermost functions in the cc-pwCVnZ basis sets. These
exponents are then added to the cc-pVnZ-PP and
cc-pwCVnZ-PP basis sets to obtain aug-cc-pVnZ-PP and
aug-cc-pwCVnZ-PP basis sets, respectively.
To complement the all-electron DK relativistic basis sets
previously developed for the first row transition metals, similar basis sets at the TZ level have also been developed for both
the second and third row elements of this work in order to
provide unambiguous comparisons with the PP-based basis
set results. In the case of Cd, a QZ quality basis set was also
optimized. The only differences between the construction of
the all-electron sets and the PP basis sets outlined above was
286
in the larger sizes of the underlying HF primitive sets and
the use of the one-electron (second-order) DKH Hamiltonian for the treatment of scalar relativity. In the case of Ag
and Cd, the spd primitive sets optimized for the cc-pVTZDK basis sets corresponded to (26s21p14d), while the ccpVQZ-DK basis for Cd consisted of (32s25p18d). The HF
primitive sets for Au and Hg consisted of (32s28p19d11f )
for cc-pVTZ-DK. These choices of primitive sets resulted in
outermost exponents very similar to the PP basis sets. The
same electronic states were used in the exponent optimizations as detailed above, but the Legendre expansion method
of Petersson et al. [43] was utilized for most of the exponents instead of carrying out full optimizations. Specifically,
for each angular momentum symmetry a six-term Legendre
expansion was used for the innermost functions while the
outermost five exponents were fully optimized. This is the
same method as was used previously for the cc-pV5Z-DK
basis sets for the first row transition metal elements [12], and
initial tests on Cd in nonrelativistic calculations yielded an
optimized (26s) set of exponents very similar to the fully
optimized set of Partridge and Faegri [44]. The present procedure differed only in the case of the 11f set for Au and Hg,
where just the six-term Legendre expansion was used. Lastly,
in each case final valence-only cc-pVnZ-DK, core-valence
cc-pwCVnZ-DK, and diffuse augmented aug-cc-pVnZ-DK
(and aug-cc-pwCVnZ-DK) basis sets (n = T for all, n = Q also
for Cd) were constructed from these primitive sets in an identical manner as the PP-based sets. Their final compositions
(cc-pVnZ-DK only, the others are analogous to the PP-based
ones) are shown in Table 1.
2.2 Benchmark calculations
The efficacy of the basis sets of the present work have been
evaluated in both atomic and molecular benchmark
calculations at the coupled cluster singles and doubles with
perturbative triples [CCSD(T)] level of theory [45,46]. Both
valence only and valence plus outer-core electron correlation calculations have been carried out. In the case of the
atomic calculations, the properties considered were the electron affinities and s 1 d 10 → s 2 d 9 excitation energies for the
coinage metals, as well as the ionization potentials of all six
elements of the present work. The open-shell atomic calculations utilized the R/UCCSD(T) variant [47–50], i.e., restricted open-shell HF (ROHF) orbitals were employed but
some spin contamination is allowed in the solution of the
CCSD equations. Molecular benchmark calculations on the
strongly bound coinage metal dimers (Cu2 , Ag2 , Au2 ) involved computing 7–9 points on their near-equilibrium potential energy curves, fitting the resulting energies to sixth
order polynomials in bond displacement coordinates, and
carrying out the usual Dunham analysis [57] to determine
equilibrium bond lengths, harmonic frequencies, anharmonicities, and dissociation energies. The group 12 van der
Waals dimers were treated in a similar manner, except that
the function counterpoise (CP) correction [52] was included.
K.A. Peterson, C. Puzzarini
Also in these latter cases, in addition to the standard diffuse
augmented basis sets doubly-augmented sets [53] (d-augcc-pVnZ) were also employed, where the additional diffuse
functions (one for each angular momentum symmetry) were
obtained by simply scaling the outermost exponents of the
aug-cc-pVnZ sets (PP and DK) by a factor of 0.40.
To thoroughly test the accuracy of the underlying pseudopotentials in the molecular calculations, spin–orbit (SO) calculations were also carried out for each of the molecules
of this study. These SO calculations employed the aug-ccpVTZ-PP basis sets in each case and utilized the SO parameters that accompany the PPs [26]. At each point on the
diatomic potential energy curves, a SO correction to the energy was obtained as the difference between the energy obtained in a CI calculation without the SO interaction and one
with it included. These calculations were carried out with the
COLUMBUS program [54], and the SO CI calculations utilized the double-group SO–CI program of Pitzer and coworkers [55], where in the present cases all possible singlet and
triplet states were mixed via the SO operator. The SO–CI
calculations on the coinage metal dimers were carried out at
the multireference configuration interaction (MRCISD) level
of theory using a complete active space self-consistent field
reference space, but with only the two valence s electrons
active. The group 12 dimers were treated at the single reference CISD level of theory. In all cases, only the valence
electrons were correlated and the dissociation limits were
calculated in the supermolecule approximation with a bond
length of 50 bohr.
Finally, in the all-electron calibration calculations
employing the cc-pVnZ-DK basis sets, the accuracy of the
second-order DKH Hamiltonian was also investigated by carrying out a limited number of calculations employing both
the DK2 and DK3 Hamiltonians [56–58] as implemented
in the UTChem program [59]. These calculations were performed at each point on the potential energy curves using second order M∅ller–Plesset perturbation theory (MP2, valence
correlation only), the aug-cc-pVTZ-DK basis sets, and with
unrestricted HF orbitals for the open-shell species. The differences in the two energies (DK2 and DK3) were then added to
the DK-CCSD(T) potential energy curves determined using
MOLPRO for a subsequent spectroscopic constant analysis.
Analogous calculations were also carried out in relation to
the atomic electron affinities and ionization potentials.
It should also be noted that in the all-electron (DK2) corevalence calculations on Au2 and Hg2 , the 5s electrons were
correlated together with the 5p electrons in order to be consistent with the PP-based work. This is problematic for these
species since the Au and Hg relativistic 4f orbitals lie higher
in energy than the 5s. (Note, in the atomic calculations these
orbitals appear in different symmetries and hence do not present any special problems.) In the present work, the two 5stype orbitals were simply rotated above the core 4f -type
orbitals prior to the CCSD(T) calculations.
The fundamental advantage in using a sequence of basis sets that exhibit systematic, regular convergence towards
the CBS limit is the possibility of extrapolating the results
Systematically convergent basis sets for transition metals
287
accurately and thereby removing basis set truncation errors.
As in previous work, we have generally utilized two extrapolation formulas [60–62],
En = ECBS + An−3
En = ECBS + Ae
−(n−1)
(1)
+ Be
−(n−1)2
(2)
where n is the cardinal number of the basis set (n = 2 for DZ,
n = 3 for TZ, etc.) and ECBS is the resulting estimate of the
CBS limit. Past experience has indicated that with basis sets
of the sizes used in the present work, Eq. (1) sometimes tends
to slightly overestimate the true CBS limit while Eq. (2) generally provides an underestimate. Hence the average value
obtained from these two expressions has been used in the
past as a conservative estimate of the actual CBS limit [5,
12, 63]. In the present work, two-point extrapolations using
Eq. (1) with QZ and 5Z basis sets have been used throughout
on only the CCSD(T) correlation energies. The HF energies
were not extrapolated in these cases and the 5Z HF values
were simply taken as the most accurate estimates of the HF
limits. In the cases of the group 12 van der Waals dimers,
the CP corrected HF and correlation energy contributions to
the binding energies were used. Extrapolations using Eq. (2)
were carried out only for the group 12 dimers, which showed
much slower basis set convergence with the aug-cc-pVnZPP basis sets (and hence more susceptible to overshooting
by Eq. (1)). In these cases a three-point extrapolation was
carried out on the total energies with Eq. (2) using basis sets
from TZ to 5Z. The CBS limits cited for the van der Waals
species correspond to the average of the values resulting from
Eqs. (1) and (2).
3 Results and discussion
3.1 Atomic properties
CCSD(T) results as a function of basis set are shown in
Tables 2, 3, and 4 for the atomic excitation energies, electron affinities, and ionization potentials, respectively. Smooth
convergence with respect to theVnZ-PP basis sets is observed
in all cases. Focusing first on the s 1 d 10 → s 2 d 9 excitation
energies of the coinage metals shown in Table 2, the valence electron correlated results all converge from below the
apparent CBS limit as the basis set is increased in size. As
noted in the previous all-electron work for the late transition
metals [12], the diffuse augmented basis set series yields a
more rapid convergence towards the CBS limit. Tests on the
Cu atom indicate that more than half of this improvement is
due to the diffuse f -type functions in the aug-cc-pVnZ-PP
basis sets, which points to the importance of sd correlation effects for this property. This effect appears to be less important
down the series Cu, Ag, Au. It should also be noted, however,
that nearly identical extrapolated CBS limits are obtained
for these excitation energies when either the cc-pVnZ-PP or
aug-cc-pVnZ-PP basis set series are utilized. The effects of
correlating the (n − 1) sp electrons are also shown in Table 2 using the (aug-)cc-pwCVnZ-PP basis sets. In each case
core-valence correlation decreases the state separation, particularly in the cases ofAg andAu where the excitation energy
is decreased by 5–6 kcal/mol at the CBS limit. The effect for
Cu is nearly zero at the CBS limit. On comparison to all-electron DK results using aug-cc-pwCVTZ-DK basis sets, it is
evident that the PP approximation overestimates the overall
effects of core-valence correlation (i.e., the s 2 d 9 state is overstabilized) by about 1–1.5 kcal/mol for this particular property. This is also exhibited in the final comparisons between
theory and experiment in Table 2, where the CCSD(T) values
are consistently too small by 0.6–2.1 kcal/mol. Accounting
for PP errors of the magnitude noted above brings all three
to within chemical accuracy.
These excitation energies were also calculated at the
CCSD(T) level by Figgen et al [26] in the course of evaluating their PP adjustments. Their basis sets were similar to
the present aug-cc-pVTZ-PP sets, however, they were used
completely uncontracted. On their own these results appeared
to be relatively far from experiment, but most of these errors
can now clearly be attributed to the use of a finite basis set
and not to inaccuracies in the pseudopotentials.
The calculated electron affinities for the coinage metal
elements shown in Table 3 exhibit excellent convergence with
respect to basis set (aug-series), both for valence-only correlated and in the core-valence calculations. In fact the aug-ccpVTZ-PP results are within 1.5 kcal/mol of the CBS limits. In
the case of the electron affinities, the differences between the
PP-based calculations and the all-electron DK ones are much
smaller than observed for the excitation energies, generally
only a few tenths of a kcal/mol even for the core-valence correlation effects. The latter values are again appreciable in the
cases of Ag and Au where (n − 1) sp correlation is calculated
to increase the electron affinities by 1.0 and 1.4 kcal/mol,
respectively. The final CBS limit results are within about
0.3 kcal/mol of experiment in each case, which reflects the
low intrinsic error of CCSD(T) for the electron affinities of
the coinage metal elements.
The ionization potentials (IPs) shown in Table 4 for all
the elements of the present work show excellent convergence
with basis set and only small differences between the PP and
all-electron DK approaches. Use of the diffuse augmented
series of basis sets leads to improved convergence rates as
was also noted previously in the all-electron first row transition metal study. This also seems to imply a relatively larger
importance of sd correlation in the neutral atoms. The corevalence correlation contributions to the IPs are accurately
predicted even with the cc-pwCVDZ-PP basis sets in these
cases. After extrapolating both the valence-only and CV contributions to their respective CBS limits, the CCSD(T) IPs are
all within 0.8 kcal/mol of experiment.
3.2 Molecular benchmark calculations
3.2.1 The coinage metal dimers: Cu2 , Ag2 , and Au2
Due to their relatively simple electronic structure for transition metal dimers, the coinage metal homonuclear diatomics,
288
K.A. Peterson, C. Puzzarini
Table 2 Calculated CCSD(T) excitation energies s 1 d 10 → s 2 d 9 for the group 11 atoms compared to the J -averaged experimental values
(kcal/mol)
Basis set a
Valence electrons correlated
Cu
Ag
(VDZ/wCVDZ)-PP
(VTZ/wCVTZ)-PP
(VQZ/wCVQZ)-PP
(V5Z/wCV5Z)-PP
CBS(Q5)
(aVDZ/awCVDZ)-PP
(aVTZ/awCVTZ)-PP
(aVQZ/awCVQZ)-PP
(aV5Z/awCV5Z)-PP
CBS(Q5)
(aVTZ/awCVTZ)-DK
(aVQZ/awCVQZ)-DK
CBS(Val. + CV)b
Expt.c
21.99
28.83
31.67
32.75
33.86
30.79
31.59
32.94
33.32
33.71
31.49
32.87
33.8
34.37
Au
89.40
92.34
94.60
95.21
95.85
93.37
93.51
94.94
95.37
95.82
94.33
40.75
42.13
43.70
44.14
44.59
42.22
42.80
43.90
44.21
44.54
42.70
89.5
91.58
39.5
40.23
Effect of (n − 1) sp core-valence correlation
Cu
Ag
Au
−1.37
−1.43
−0.79
−0.37
0.06
−1.90
−1.01
−0.59
−0.26
0.08
−0.10
0.33
−8.98
−8.06
−7.39
−6.88
−6.36
−8.94
−7.76
−7.29
−6.82
−6.33
−6.14
−7.75
−6.64
−6.06
−5.57
−5.06
−7.69
−6.45
−6.01
−5.54
−5.05
−4.88
a
Abbreviated as: VnZ = cc-pVnZ, wCV nZ = cc-pwCVnZ, aV nZ = aug-cc-pVnZ, awCVnZ = aug-cc-pwCVnZ (n = D, T, Q, 5). The valence
sets are used in the valence electrons correlated calculations, while the core-valence sets were used to determine the core-valence effects (valence
and valence + outer-core calculations)
b
Obtained from the valence correlated and core-valence CBS limits (aVnZ series)
c
Ref. [88]
Table 3 Calculated CCSD(T) electron affinities for the group 11 atoms compared to the experimental values (kcal/mol)
Basis set a
Valence electrons correlated
Cu
Ag
(aVDZ/awCVDZ)-PP
(aVTZ/awCVTZ)-PP
(aVQZ/awCVQZ)-PP
(aV5Z/awCV5Z)-PP
CBS(Q5)
(aVTZ/awCVTZ)-DK
(aVQZ/awCVQZ)-DK
CBS(Val. + CV)c
Expt.d
25.73
27.17
27.90
28.28
28.61
27.31
28.04
28.8
28.50
Au
26.93
28.24
28.90
29.12
29.30
28.26
48.15
49.98
50.93
51.21
51.42
50.43b
30.3
30.08
52.9
53.24
Effect of (n − 1) sp core-valence correlation
Cu
Ag
Au
0.30
0.28
0.26
0.23
0.20
0.21
0.18
1.33
1.21
1.17
1.11
1.04
0.98
2.18
1.85
1.73
1.60
1.46
1.41
a
Abbreviated as: aVnZ = aug-cc-pVnZ, awCVnZ = aug-cc-pwCVnZ (n = D, T, Q, 5). The valence sets are used in the valence electrons correlated
calculations, while the core-valence sets are used to determine the core-valence effects (valence and valence + outer-core calculations)
b
Contains a contribution of +0.40 kcal/mol for the difference between DK2 and DK3. See the text
c
Obtained from the valence correlated and core-valence CBS limits
d
Ref. [89]
especially Cu2 and Au2 , have been the subject of numerous
theoretical studies. The most extensive previous study of the
Au2 ground state was by Hess and Kaldor [64], who utilized large basis sets and the all-electron scalar relativistic
DK Hamiltonian with Møller–Plesset and CCSD(T) methods. This molecule was also recently addressed by Fleig and
Visscher [65] using a spin-free Dirac–Coulomb Hamiltonian
approach and CCSD(T). All three species have been studied recently by Hirao and coworkers [27] using the MP2
and CCSD(T) methods with the DK3 Hamiltonian. They utilized newly developed basis sets that were similar in size and
accuracy to the aug-cc-pVTZ-DK sets of the present work.
The present CCSD(T) results for the ground electronic
states of the coinage metal dimers are shown in Tables 5,
6, and 7 for Cu2 , Ag2 , and Au2 , respectively, using both the
cc-pVnZ-PP and aug-cc-pVnZ-PP basis set series for
valence-only correlation and the cc-pwCVnZ-PP sets for the
calculation of core-valence correlation effects. For all three
molecules the convergence rate of the spectroscopic constants is noticeably improved when the diffuse augmented
sets are used, which would appear to reflect the importance
of d–d dispersion interactions. The resulting CBS limits are
nearly the same, however, regardless of which basis set series
is used.
The copper dimer exhibits the largest difference between
the cc-pVnZ-PP and aug-cc-pVnZ-PP derived CBS limits,
where the latter CBS limit for De is larger than the cc-pV
nZ-PP extrapolated value by 0.76 kcal/mol. The calculated
core-valence correlation effects for Cu2 are essentially zero,
which agrees well with our previous all-electron study [12].
The effects of SO coupling are calculated to be completely
negligible to the number of digits reported in Table 5. Upon
comparing with the experiment, the final calculated spectroscopic constants for Cu2 agree very well, with the equilibrium
distance being a bit too short (−0.005 Å) and the dissociation
energy being slightly underestimated by about 0.6 kcal/mol.
Systematically convergent basis sets for transition metals
289
Table 4 Calculated CCSD(T) ionization potentials for the group 11 and 12 atoms compared to the J -averaged experimental values (kcal/mol)
Basis set
Valence-only
cc-pVDZ-PP
cc-pVTZ-PP
cc-pVQZ-PP
cc-pV5Z-PP
CBS(Q5)
cc-pVTZ-DK
Effect of (n − 1) sp
correlation
cc-pwCVDZ-PP
cc-pwCVTZ-PP
cc-pwCVQZ-PP
cc-pwCV5Z-PP
CBS(Q5)
cc-pwCVTZ-DK
CBS(Val. + CV)b
Expt.c
a
b
c
Cu
Ag
Au
Zn
Cd
Hg
165.26
171.89
174.89
176.47
178.13
171.94
163.05
168.06
170.69
171.88
173.13
168.56
200.23
206.05
208.08
209.08
210.13
206.34a
211.18
212.71
214.50
215.42
216.71
212.40
200.71
202.01
203.97
204.83
206.07
202.20
231.33
234.78
236.52
237.33
238.45
234.50a
0.66
0.78
0.76
0.76
0.75
0.66
178.9
178.17
2.55
2.40
2.40
2.36
2.32
1.97
175.5
174.71
2.92
2.66
2.63
2.53
2.42
2.11
212.6
212.75
0.12
0.33
0.41
0.46
0.50
0.25
217.2
216.63
1.51
1.51
1.60
1.61
1.62
1.25
207.7
207.40
2.11
2.03
2.06
1.99
1.92
1.53
240.4
240.69
These contain contributions of +0.63 (Au) and +0.62 (Hg) kcal/mol for the differences between DK2 and DK3. See the text
Obtained from the valence correlated and core-valence CBS limits
Ref. [88,90]
Table 5 Calculated CCSD(T) spectroscopic properties of X 1 g+ Cu2 as a function of basis set and compared to experiment
Valence only
CVb
SO
CBS(aQ5) + CV + SO
Expt.c
a
b
c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (kcal/mol)
cc-pVDZ-PP
cc-pVTZ-PP
cc-pVQZ-PP
cc-pV5Z-PP
CBS(Q5)a
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBS(aQ5)a
cc-pwCVDZ-PP
cc-pwCVTZ-PP
cc-pwCVQZ-PP
cc-pwCV5Z-PP
CBS(wQ5)a
aug-cc-pVTZ-PP
2.2375
2.2319
2.2251
2.2186
2.2147
2.2357
2.2213
2.2166
2.2154
2.2146
0.0001
0.0007
−0.0001
−0.0004
−0.0007
0.0000
2.214
2.2193
264.8
260.5
263.3
266.1
268.8
258.1
266.1
266.7
267.7
268.6
2.5
1.5
1.0
1.3
1.5
0.0
270.1
266.46
1.13
1.11
0.99
1.02
1.04
0.92
1.03
1.02
1.02
1.02
0.04
0.04
−0.01
0.01
0.02
0.00
1.04
1.04
45.21
43.69
44.55
45.35
45.86
44.41
45.39
46.02
46.36
46.62
0.33
−0.02
−0.08
−0.06
−0.04
0.00
46.6
47.93±0.57
CBS extrapolation of the CCSD(T) correlation energy based on Eq. (1)
Effects of correlating the (n − 1) sp electrons
Refs. [91,92]
The final calculated harmonic frequency is larger than the
experimental value by ∼4 cm−1 , which is consistent with
the slightly too short re . These results are nearly identical to
the all-electron DK relativistic values of Ref. [12] that were
otherwise calculated at the same level of theory.
In the case of the silver dimer (Table 6), the core-valence
correlation effects are much larger than Cu2 , shortening re
by 0.011 Å, increasing ωe by 4.7 cm−1 , and increasing De
by 0.9 kcal/mol at the CBS limit. The effect of SO coupling
on the spectroscopic constants is very small, but larger than
in Cu2 , i.e., re is decreased by 0.001 Å and De increased by
0.07 kcal/mol. The final CCSD(T) results shown in Table 6
when basis set incompleteness, core-valence correlation, and
SO coupling are taken into account agree well with the available experimental values.As in Cu2 , the bond length is slightly
underestimated in the CCSD(T) valence + CV CBS limit
(+SO) (−0.008 Å) and the harmonic frequency overestimated
(+4 cm−1 ), but in the case of Ag2 the calculated dissociation energy is actually a bit higher than the experimental
value [66] and slightly outside of the stated experimental
uncertainty. As discussed below, part of these differences
can be attributed to small inaccuracies in the PP treatment
with the remaining error being predominately intrinsic to the
CCSD(T) method.
The present results for Au2 shown in Table 7 exhibit
similar trends as the other coinage metal dimers. In this case
290
K.A. Peterson, C. Puzzarini
Table 6 Calculated CCSD(T) spectroscopic properties of X 1 g+ Ag2 as a function of basis set and compared to experiment
Valence Only
CVb
SO
CBS(aQ5) + CV + SO
Expt.c
a
b
c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (kcal/mol)
cc-pVDZ-PP
cc-pVTZ-PP
cc-pVQZ-PP
cc-pV5Z-PP
CBS(Q5)a
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBS(aQ5)a
cc-pwCVDZ-PP
cc-pwCVTZ-PP
cc-pwCVQZ-PP
cc-pwCV5Z-PP
CBS(wQ5)a
aug-cc-pVTZ-PP
2.5866
2.5575
2.5440
2.5391
2.5351
2.5675
2.5454
2.5377
2.5351
2.5339
−0.0113
−0.0110
−0.0113
−0.0111
−0.0108
−0.0012
2.522
2.5303
176.9
185.3
188.0
190.5
192.9
182.7
189.2
190.9
191.2
191.5
5.2
4.9
4.8
4.8
4.7
0.2
196.4
192.40
0.54
0.65
0.62
0.60
0.57
0.59
0.61
0.62
0.62
0.62
−0.01
0.00
0.00
0.02
0.04
0.00
0.66
0.64
34.25
35.33
36.42
37.30
38.10
35.79
37.30
38.03
38.28
38.45
1.41
1.00
0.95
0.93
0.91
0.07
39.4
38.0 ± 0.8
CBS extrapolation of the CCSD(T) correlation energy based on Eq. (1)
Effects of correlating the (n − 1) sp electrons
Refs. [66,93,94]
Table 7 Calculated CCSD(T) spectroscopic properties of X 1 g+ Au2 as a function of basis set and compared to experiment
Valece only
CVb
SO
CBS(aQ5) + CV + SO
Expt.c
a
b
c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (kcal/mol)
cc-pVDZ-PP
cc-pVTZ-PP
cc-pVQZ-PP
cc-pV5Z-PP
CBS(Q5)a
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBS(aQ5)a
cc-pwCVDZ-PP
cc-pwCVTZ-PP
cc-pwCVQZ-PP
cc-pwCV5Z-PP
CBS(wQ5)a)
aug-cc-pVTZ-PP
2.5253
2.5093
2.4972
2.4936
2.4915
2.5203
2.5002
2.4946
2.4903
2.4876
−0.0119
−0.0106
−0.0104
−0.0106
−0.0107
−0.0049
2.472
2.4715
175.5
181.6
183.7
185.5
187.1
180.2
184.1
185.6
186.3
186.9
4.9
4.0
3.8
3.6
3.5
1.1
191.5
191.05
0.40
0.41
0.42
0.41
0.41
0.43
0.41
0.41
0.42
0.42
−0.01
−0.01
0.00
0.00
0.00
0.00
0.42
0.42
45.81
47.30
48.94
50.04
50.95
47.39
49.50
50.52
50.90
51.15
2.60
1.80
1.61
1.53
1.45
0.62
53.2
53.5±0.1
CBS extrapolation of the CCSD(T) correlation energy based on Eq. (1)
Effects of correlating the (n − 1) sp electrons
Refs. [94,96]
while the CBS limit dissociation energy is nearly the same
regardless of which basis set series is used (cc-pVnZ-PP versus aug-cc-pVnZ-PP), the CBS limit bond length is about
0.004 Å shorter when the diffuse augmented series is used.
The core-valence effects are very similar to those calculated
for Ag2 with a somewhat larger increase in De , 1.45 kcal/mol
at the CBS limit. These CV effects are very similar to the
counterpoise-corrected DK-CCSD(T) results reported previously by Hess and Kaldor [64] (for 5p5d6s correlation).
In addition, the SO effects on the spectroscopic constants
shown in Table 7, −0.0049 Å for re , 1.1 cm−1 for ωe , and
0.62 kcal/mol for De , are very similar to those obtained previously by van Lenthe et al. [67] (ZORA-GGA) and Lee
et al. [68] [REP-CCSD(T)], however the SO effect on De
in this work is about 50% less than their values. The final CCSD(T)/CBS+CV+SO spectroscopic properties shown
in Table 7 are in excellent agreement with the experimental
values.
Table 8 shows the results of all-electron DK2 and DK3
calculations with explicit comparisons to the analogous PPbased results. On the one hand, use of the DK3 Hamiltonian
is observed to yield nearly identical spectroscopic properties as those obtained at the DK2 level. The effect is completely negligible for Cu2 and barely discernable for Ag2 .
In the case of Au2 , the differences between the DK3 and
DK2 values are just −0.0025 Å for re and +0.25 kcal/mol for
Systematically convergent basis sets for transition metals
291
Table 8 Comparison of CCSD(T) all-electron DK and PP-based spectroscopic constants for Cu2 , Ag2 , and Au2
Valence-only
Cu2
Ag2
Au2
Val. + (n − 1) sp
Cu2
Ag2
Au2
Basisa
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (kcal/mol)
aug-cc-pwCVTZ-DK
DK3b
PP(val)c
aug-cc-pwCVTZ-DK
DK3b
PP(val)c
aug-cc-pwCVTZ-DK
DK3b
PP(val)c
2.2230
−0.0001
0.0021
2.5472
−0.0005
0.0033
2.4977
−0.0025
−0.0016
266.0
0.0
0.0
188.2
0.1
−0.3
184.4
0.6
0.7
1.04
0.00
0.02
0.58
0.00
−0.02
0.43
−0.02
−0.01
45.27
0.01
−0.09
37.06
0.02
−0.17
49.63
0.25
0.22
aug-cc-pwCVTZ-DK
PP(CV)d
aug-cc-pwCVTZ-DK
PP(CV)d
aug-cc-pwCVTZ-DK
PP(CV)d
2.2233
0.0010
2.5380
0.0031
2.4898
0.0041
267.1
−0.9
192.4
−0.9
187.4
−1.3
1.04
−0.03
0.61
0.03
0.43
0.01
45.17
−0.16
37.87
−0.33
50.93
−0.72
a
The results using aug-cc-pwCV nZ-DK basis sets corresponded to all-electron calculations employing the second-order DKH Hamiltonian
Effects due to using the DK3 Hamiltonian compared to DK2. See the text
c
Difference between the all-electron DK3 (DK2 + DK3) and PP-based results for valence-only correlation
d
Difference between the core-valence effect calculated with the all-electron DK2 method and those derived from PP calculations
b
De (DK3 – DK2 in each case). Of course the total energy
is much lower in the DK3 calculations, −41 Eh compared
to DK2, but this is obviously of little practical importance
for the properties of interest in this study. A limited number
of tests on Au2 comparing the DK3 and DK2 treatments of
scalar relativity for core-valence correlation effects yielded
completely negligible differences between the two. Using the
CCSD(T) method also yielded results essentially identical to
the MP2 values. As shown in Table 8, the differences between
DK- and PP-based treatments for valence-only correlation,
PP(val), are relatively small, exhibiting maximum differences of just 0.003 Å for re (Ag2 ) and 0.2 kcal/mol for De
(Ag2 and Au2 ). For these strongly bound molecules, these
results are expected to be stable with respect to further basis
set extensions. Similar differences are also calculated for the
core-valence correlation effects, where in general the PPbased calculations slightly overestimate the effects of correlating the (n − 1) sp electrons. Assuming additivity, the final
CBS limit PP results of Tables 5–7 could be modified by the
sum of the PP(val) and PP(CV) values given in Table 8
to approximate all-electron CBS limit DK3 results (+SO). In
particular, this would improve the agreement with experiment
for Ag2 , especially for the equilibrium bond length.
It should also be mentioned that preliminary versions of
the present basis sets have been used in benchmark-quality
calculations on other strongly bound molecules with similar
results in regards to accuracy as the present work [5,69–71].
This is particularly evident in our previous calculations on
the ground states of the diatomic gold halides [70], (AuF,
AuCl, AuBr, and AuI), where nearly perfect agreement with
experiment was obtained after basis set extrapolation and
inclusion of both core-valence correlation and SO coupling
effects.
3.2.2 The group 12 van der Waals dimers: Zn2 , Cd2 ,
and Hg2
The group 12 dimers have been the subject of numerous
theoretical and experimental studies due to their possible use
as excimer laser substances. The ground states are only very
weakly bound and the accuracy of the calculated spectroscopic properties are expected to show a strong dependence
on the basis sets employed. The present results for Zn2 , Cd2 ,
and Hg2 are shown in Tables 9, 10, and 11, respectively, where
they are also compared to the available experimental values.
In general, while the singly augmented (aug-cc-pVnZ-PP)
and doubly augmented (d-aug-cc-pVnZ-PP) basis sets both
lead to nearly identical CBS limits in each case, the d-aug
series converge much more rapidly and the resulting spread
between the CBS limits obtained with Eqs. (1) and (2) is
much smaller.
Focusing first on the Zn dimer results in Table 9, a very
strong dependence of the equilibrium bond length on basis
set is observed. Even with the d-aug series of basis sets,
extrapolation to the CBS limit leads to a decrease from the
d-aug-cc-pV5Z-PP value of slightly more than 0.02 Å. The
analogous difference between the CBS and aug-cc-pV5ZPP values is 0.07 Å, but an identical CBS limit is obtained
(with a much larger estimated uncertainty). This would appear to demonstrate the robustness of the present basis set
extrapolation procedure. The two CBS limits for De are also
within their estimated uncertainties. Core-valence correlation effects are calculated to have only a small effect on the
derived spectroscopic properties of Zn2 , i.e., increasing re by
0.007 Å and decreasing De by just 3 cm−1 . The core-valence
correlation effects on re and De calculated by Yu and Dolg
[72], however, at the MP4 level of theory would appear to be
292
K.A. Peterson, C. Puzzarini
Table 9 Calculated counterpoise correcteda CCSD(T) spectroscopic properties of X 1 g+ Zn2 as a function of basis set and compared to experiment
Valence only
CV
SO
CBS(daVnZ) + CV + SO
Expt.c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (cm−1 )
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBSb
d-aug-cc-pVDZ-PP
d-aug-cc-pVTZ-PP
d-aug-cc-pVQZ-PP
d-aug-cc-pV5Z-PP
CBSb
aug-cc-pwCVDZ-PP
aug-cc-pwCVTZ-PP
aug-cc-pwCVQZ-PP
aug-cc-pVTZ-PP
4.3536
4.1932
4.0148
3.9132
3.840 [21]
4.2048
3.9538
3.8910
3.8619
3.840 [6]
0.0075
0.0071
0.0068
−0.0002
3.847
4.19
16.4
17.6
19.6
21.9
23.9 [6]
17.6
21.1
22.0
23.1
24.0 [2]
−0.1
−0.1
−0.1
0.0
23.9
25.9±0.2
0.57
0.54
0.51
0.56
0.59 [0]
0.51
0.54
0.49
0.54
0.56 [1]
0.03
0.04
0.00
−0.02
0.54
0.60±0.05
111.6 (101)
131.3 (37)
163.0 (27)
193.0 (13)
221 [8]
135.4 (120)
183.7 (42)
205.8 (39)
218.1 (26)
229 [3]
−2.0
−2.5
−2.9
0.0
226
279.1
a
The BSSE removed by the counterpoise correction is given in parentheses for De
Obtained as the average of Q5 extrapolations using Eq. (1) and TQ5 extrapolations using Eq. (2). The estimated uncertainty in the extrapolation
is given in square brackets for the last digit
c
Ref. [73]
b
Table 10 Calculated counterpoise correcteda CCSD(T) spectroscopic properties of X 1 g+ Cd2 as a function of basis set and compared to
experiment
Valence only
CV
SO
CBS(daVnZ) + CV + SO
Expt.c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (cm−1 )
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBSb
d-aug-cc-pVDZ-PP
d-aug-cc-pVTZ-PP
d-aug-cc-pVQZ-PP
d-aug-cc-pV5Z-PP
CBSb
aug-cc-pwCVDZ-PP
aug-cc-pwCVTZ-PP
aug-cc-pwCVQZ-PP
aug-cc-pVTZ-PP
4.4438
4.1760
4.0002
3.9366
3.889[14]
4.1794
3.9864
3.9318
3.9256
3.921[1]
−0.0212
−0.0205
−0.0239
−0.0024
3.894
4.07
13.3
15.9
18.6
20.2
21.5[4]
16.8
19.3
20.5
20.2
20.0[1]
0.0
0.1
0.2
0.0
20.2
23.0
0.24
0.28
0.27
0.31
0.28[2]
0.28
0.27
0.32
0.24
0.19[1]
0.00
−0.01
0.00
0.01
0.20
0.40 ± 0.01
152.4 (170)
202.3 (48)
261.4 (37)
296.5 (17)
327[9]
219.6 (205)
282.5 (74)
307.6 (61)
315.1 (32)
321[2]
0.1
1.5
3.4
0.7
325
330.5
a
The BSSE removed by the counterpoise correction is given in parentheses for De
Obtained as the average of Q5 extrapolations using Eq. (1) and TQ5 extrapolations using Eq. (2). The estimated uncertainty in the extrapolation
is given in square brackets for the last digit
c
Ref. [73]
b
strongly overestimated compared to these values. The inaccuracy of MP4 for this purpose was also noted in that same study
based on additional CCSD(T) calculations. As also shown in
Table 9, SO coupling effects are negligible for the Zn dimer.
Our final CCSD(T)/CBS+CV+SO results for the re and
De of Zn2 are in relatively poor agreement with the
experimental values [72]. The experimental re , however, does
not correspond to a directly measured quantity, but was derived based on modeling Zn2 as a pure van der Waals molecule. As previously discussed by Yu and Dolg [72], however,
all of the group 12 dimers have significant covalent contributions to their binding that would lead to a shortening of the
bond length compared to a van der Waals interaction alone.
Recently the experimental equilibrium bond lengths of the
group 12 dimers were re-evaluated by combining spectroscopic data and viscosity measurements [74]. The resulting
value for Zn2 (4.62 ± 0.05 Å), however, is longer than the
previously cited experimental re and hence even further from
the present predicted value. Furthermore, the predicted De of
this work, 226 cm−1 , is more than 50 cm−1 below the experimental value. This is well outside the expected accuracy
of the present calculations. The origin of this difference is
not known at this time. Our results, however, are consistent
with previous theoretical studies, notably the accurate calculations of Yu and Dolg [72] and the recent calculations of
Ellingsen et al. [75]. The present work is the first to determine
Systematically convergent basis sets for transition metals
293
Table 11 Calculated counterpoise correcteda CCSD(T) spectroscopic properties of X 1 g+ Hg2 as a function of basis set and compared to
experiment
Valence only
CV
SO
CBS(daVnZ) + CV + SO
Expt.c
Basis
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (cm−1 )
aug-cc-pVDZ-PP
aug-cc-pVTZ-PP
aug-cc-pVQZ-PP
aug-cc-pV5Z-PP
CBSb
d-aug-cc-pVDZ-PP
d-aug-cc-pVTZ-PP
d-aug-cc-pVQZ-PP
d-aug-cc-pV5Z-PP
CBSb
aug-cc-pwCVDZ-PP
aug-cc-pwCVTZ-PP
aug-cc-pwCVQZ-PP
aug-cc-pVTZ-PP
4.2200
3.9367
3.8169
3.7689
3.733 [10]
4.0623
3.8368
3.7858
3.7530
3.725 [7]
−0.0377
−0.0344
−0.0385
−0.0217
3.665
3.63 ± 0.04
12.0
15.7
17.2
18.4
19.4 [3]
14.8
17.5
18.4
18.5
18.7 [1]
0.4
0.5
0.6
0.2
19.5
19.6 ± 0.3
0.23
0.21
0.21
0.22
0.22 [0]
0.22
0.22
0.21
0.21
0.23 [0]
−0.01
0.01
0.01
0.00
0.24
0.26 ± 0.05
163.3 (217)
263.7 (70)
316.9 (34)
349.9 (16)
378 [8]
228.9 (262)
317.4 (107)
350.1 (58)
361.5 (28)
371 [3]
8.3
13.1
17.0
10.0
398
380 ± 15
a
The BSSE removed by the counterpoise correction is given in parentheses for De
Obtained as the average of Q5 extrapolations using Eq. (1) and TQ5 extrapolations using Eq. (2). The estimated uncertainty in the extrapolation
is given in square brackets for the last digit
c
Value of re from Ref. [77], ωe and ωe xe from Ref. [79], De from Ref. [78]. See Ref. [79] for other experimental values
b
CBS limits for these spectroscopic constants, and the present
CBS+CV+SO values should be the most reliable to date for
re , and perhaps even De .
The comparison of the present PP-based results and
all-electron DK calculations are shown in Table 12 using
d-aug QZ basis sets. As with the Cu2 molecule, extensions
of the DK Hamiltonian from second- to third-order leads to
completely negligible differences in spectroscopic constants
for Zn2 . By combining the valence-only and core-valence
DK2–PP differences, PP(val)+PP(CV), the resulting
estimated error from the pseudopotential approximation is
also determined to be negligible in the present case.
The results for the Cd dimer shown in Table 10 exhibit
similar trends as in Zn2 , although in general even more rapid
convergence towards the CBS limit is observed with the daug series of basis sets as compared to Zn2 . This results in
CBS limits for re that differ by about 0.03 Å depending on
which basis set series (aug or d-aug) is used. The value obtained using the d-aug-cc-pVnZ-PP sets is certainly the most
reliable. Contrastingly, the CBS limits for De are very similar between the two series, much like the case of Zn2 . Corevalence correlation effects are calculated to be relatively large
for Cd2 in regards to the bond length, i.e., −0.024 Å with the
aug-cc-pwCVQZ-PP basis set, but they only lead to a modest
increase in De with this same basis (+3.4 cm−1 ). The inclusion of SO coupling slightly decreases re (−0.002 Å) and
essentially leaves De unchanged. The final CBS+CV+SO
values shown in Table 10 are in excellent agreement with
the experimental dissociation energy [73], but the predicted
re of this work (3.894 Å) is smaller than in the experiment.
This latter value, however, was also modeled as a van der
Waals molecule as in Zn2 and is expected to be somewhat
too long. The recently revised experimental value that incorporates viscosity measurements [74] is also much too long
(4.33 ± 0.05 Å). The best previous ab initio results were the
CCSD(T) values of Yu and Dolg [72], and these were similar
in quality to our current aug-cc-pV5Z-PP values. The comparisons between PP and DK calculations shown in Table 12
show the same trends as Zn2 . Both the valence-only and CV
PP values are of similar magnitude but opposite in sign, so
that the total PP errors again appear to be negligible in this
case.
The Hg dimer is the most extensively studied member of
this series, both by theory and experiment. In particular, high
resolution spectroscopy experiments [76–80] have resulted in
accurate molecular parameters for the ground state. Recent
ab initio studies include the work of Schwerdtfeger et al.
[81,82], Munro et al. [83] and Dolg and Flad. [84]. In each
of these cases the quasirelativistic PP of Häussermann et al.
[32] was used with relatively large one-particle basis sets in
order to minimize the basis set superposition error (BSSE).
For example, the basis set used in the most recent work of
Schwerdtfeger et al. [81] was optimized [85] at the MP2 level
of theory and included up to h-type functions. In addition,
Kunz et al. [86] reported all-electron calculations on Hg2
employing the second-order DKH Hamiltonian. In each of
these studies, high-level electron correlation methods were
employed such as CCSD(T) and good agreement with the
experiment was generally obtained, but the basis sets used
did not allow an extrapolation to the CBS limit, which limited the accuracy attainable in these studies.
The results of the present work are shown in Table 11.
The convergence of the counterpoise corrected spectroscopic
properties is very regular with the d-aug-cc-pVnZ-PP series exhibiting the most rapid convergence with increasing n.
Core-valence correlation effects are substantial, decreasing
the bond length by 0.039 Å and increasing De by 17 cm−1 .
Previous workers [82–84] had also investigated the effects
294
K.A. Peterson, C. Puzzarini
Table 12 Comparison of CP-corrected CCSD(T) all-electron DK and PP-based spectroscopic constants for Zn2 , Cd2 , and Hg2
Valence-only
Zn2
Cd2
Hg2
Val. + (n − 1) sp
Zn2
Cd2
Hg2
Basisa
re (Å)
ωe (cm−1 )
ωe xe (cm−1 )
De (cm−1 )
d-aug-cc-pVQZ-DK
DK3b
PP(val)c
d-aug-cc-pVQZ-DK
DK3b
PP(val)c
d-aug-cc-pVTZ-DK
DK3b
PP(val)c
3.8887
0.0000
−0.0023
3.9259
0.0000
−0.0059
3.8275
−0.0028
−0.0121
22.1
0.0
0.1
20.3
0.0
−0.2
17.1
0.0
−0.4
0.51
0.00
0.02
0.29
0.00
−0.03
0.19
0.00
−0.03
206.6
0.0
0.8
309.2
−0.2
1.4
319.6
−0.7
1.4
3.9937
0.0018
3.9573
0.0061
3.8909
0.0075
20.0
0.0
19.4
0.0
16.2
−0.1
0.37
0.00
0.32
0.01
0.15
−0.01
166.1
0.1
280.2
−0.6
279.0
−2.4
aug-cc-pwCVQZ-DK
PP(CV)d
aug-cc-pwCVQZ-DK
PP(CV)d
aug-cc-pwCVTZ-DK
PP(CV)d
a
The results using aug-cc-pwCVnZ-DK basis sets corresponded to all-electron calculations employing the second-order DKH Hamiltonian
Effects due to using the DK3 Hamiltonian compared to DK2. See the text
c
Difference between the all-electron DK3 (DK2 + DK3) and PP-based results for valence-only correlation
d
Difference between the core-valence effect calculated with the all-electron DK2 method and those derived from PP calculations
b
of correlating the 5s and 5p electrons, but their results for
the bond shortening due to core-valence correlation were all
much too large, −0.1 Å at the MP4 level to −0.07 Å at the
MP2 level. These results can mainly be attributed to using
MPn methods for these systems, which are now known to
strongly overestimate correlation effects in these cases. The
CCSD(T) values of Yu and Dolg [72] (re = −0.074 Å and
De = 32 cm−1 ), while smaller than the MP4 results, are
too large by about a factor of two, perhaps due to basis set
effects. As shown in Table 11, SO coupling is estimated to
decrease re by an additional 0.022 Å with an increase in De by
10 cm−1 . These values are similar to those obtained earlier by
Dolg and Flad [84] using a two-electron PP (re = −0.017,
De = +8 cm−1 ), but smaller by about a factor of two from
their 20-electron PP results. The CBS+CV+SO results shown
in Table 11 should represent the most accurate theoretical
values to date for this molecule. The predicted equilibrium
bond length of 3.665 Å from this work is shorter than the
experimental value obtained by simulations of the G ← X
electronic spectrum by Koperski et al. [80] 3.69±0.01 Å, but
within the stated uncertainty of the directly determined value
of 3.63 ± 0.04 Å by van Zee et al. [77]. Curiously the experimentally derived value of Ref. [73], 3.66±0.05 Å, is nearly
identical to our predicted result for re . This is in strong contrast to their results for Zn2 and Cd2 as discussed above. In
the case of De , as shown in Table 11 the predicted equilibrium binding energy from this work is just at the upper end
of the experimental range.
Comparison of the PP-based results for Hg2 with the corresponding DK values are given in Table 12. As observed
previously for Au2 , the differences between results calculated with the DK3 and DK2 Hamiltonians are nearly negligible, i.e., differences of 0.0028 Å in re and 0.7 cm−1 in
De . Using basis sets of triple-ζ quality in the DK calculations, the valence-only result for re is slightly shorter in the
all-electron calculations compared to the PP values, while
the binding energy is within 2 cm−1 . As observed for Zn2
and Cd2 , the difference between the core-valence correlation
effects determined in the DK and PP calculations are of similar magnitude as PP(val), but opposite in sign. Hence the
total apparent errors in the PP approximation, defined as the
differences between the PP and DK3 results, are essentially
negligible for all the spectroscopic constants evaluated for
Hg2 in this study.
Lastly, it should be particularly noted that the same pseudopotentials used in this work have been previously applied
to the group 12 dimers by Figgen et al. [26] in benchmark
CCSD(T) calculations. The basis sets they used in their study
yielded spectroscopic properties intermediate between the
present valence correlated d-aug-cc-pVTZ-PP and d-aug-ccpVQZ-PP values. As suggested in their work and unambiguously confirmed in the present study by explicit basis set
extrapolations and all-electron relativistic calculations, the
resulting errors compared to the experiment that they observed should not be attributed to errors in the PP adjustment
but to residual basis set truncation errors and incomplete electron correlation recovery.
4 Conclusions
Series of correlation consistent basis sets have been developed
for the groups 11 and 12 elements based on recently reported
relativistic pseudopotentials. Basis sets for both valence-only
and outer-core electron correlation are reported, as well as
these basis sets extended by diffuse functions. A limited number of relativistic all-electron basis sets of triple-ζ quality
(and quadruple-ζ for Cd) were also optimized for each element of this study to provide accurate comparisons to the
PP-based results. Large-scale atomic and molecular bench-
Systematically convergent basis sets for transition metals
mark calculations were carried out at the CCSD(T) level of
theory. Smooth, regular convergence towards the complete
basis set limit was generally obtained when series of the new
basis sets were used. This allowed for a systematic removal
of basis set truncation errors, which was shown to be very
important for approaching the experimental values as well
as assessing the accuracy of the pseudopotential approximation. In general, excellent agreement with experiment was
observed where available, and in the case of the group 12 van
der Waals dimers, predictions for their equilibrium geometries were made in each case that are expected to be more
accurate than the existing experimental results. Work is currently underway to extend these basis sets to the remaining
elements of the transition metal block [33].
All of the basis sets presented in this work will be made
available for download from the Pacific Northwest National
Laboratory (PNNL) basis set website [87], as well as by request from one of the authors (KAP).
Acknowledgements This work was supported by the National Science
Foundation under CHE-0111282. Partial support by the Division of
Chemical Sciences in the Office of Basis Energy Sciences of the U.S.
Department of Energy (DOE) is also gratefully acknowledged. Cristina
Puzzarini was partially supported by the University of Bologna under
the “Marco Polo” project. The authors would like to thank Profs. Stoll
(Stuttgart) and Dolg (Köln) for generously providing their pseudopotentials prior to publication.
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